arXiv research

Guarantees uniform convergence for square-root Lipschitz losses.

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $α:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(α(x))_k=|<x,f_k>|^2$, where $…

Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.

The problem of optimizing unknown costly-to-evaluate functions has been studied for a long time in the context of Bayesian Optimization. Algorithms in this field aim to find the optimizer of the function by asking only a few function evaluations at locations carefully selected based on a posterior model. In this paper,…

New algorithm tackles delayed feedback in Lipschitz bandits with sublinear regret.

Global stability bounds for matrix frames in phase retrieval problems.

A major issue in harmonic analysis is to capture the phase dependence of frequency representations, which carries important signal properties. It seems that convolutional neural networks have found a way. Over time-series and images, convolutional networks often learn a first layer of filters which are well localized i…

Study on free boundary problems in RCD spaces, proving existence and regularity.

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…

In this work we compute lower Lipschitz bounds of $\ell_p$ pooling operators for $p=1, 2, \infty$ as well as $\ell_p$ pooling operators preceded by half-rectification layers. These give sufficient conditions for the design of invertible neural network layers. Numerical experiments on MNIST and image patches confirm tha…

The covariance of a stationary process $X$ is diagonalized by a Fourier transform. It does not take into account the complex Fourier phase and defines Gaussian maximum entropy models. We introduce a general family of phase harmonic covariance moments, which rely on complex phases to capture non-Gaussian properties. The…

Markov chain decoders improve generative models' ability to produce heavy-tailed data.

We prove longtime existence and estimates for solutions to a fully nonlinear Lagrangian parabolic equation with locally $C^{1,1}$ initial data $u_0$ satisfying either (1) $-(1+η) I_n\leq D^2u_0 \leq (1+η)I_n$ for some positive dimensional constant $η$, (2) $u_0$ is weakly convex everywhere or (3) $u_0$ satisfies a larg…

Paper analyzes latent space geometry in generative models using Fisher information.

New algorithm helps escape saddle points in optimization problems.

We tackle tensor denoising with unknown permutations, achieving optimal recovery with polynomial estimators.

Study phase retrieval under misspecified models using generative priors.

In this paper we show that two Lagrangian graphs over the torus in $\mathbb{C}^n$ with large Lagrangian phase can be connected via Lipschitz continuous geodesic with respect to the $L^2$ metric on the space of Lagrangian submanifolds. In particular, the geodesic for Lagrangian graphs over the torus in $\mathbb{C}^n$ ca…

Near-optimal sample complexity for phase retrieval with generative priors.

Investigates fast prediction rates with limited expert advice.

For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifold…

The paper develops a new method for estimating non-parametric regression functions with spatio-temporal dependencies.

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $Δu = f(u)$ on hyperbolic space $\HH^n$ must reduce to functions of one variable provided they admit asympto…

Consistent partial identification of causal effects proved for neural models.

Paper proposes a new method for PCA using generative models.

New method generates private synthetic data with optimal utility for smooth queries.

Enhances neural networks' robustness against adversarial samples without sacrificing clean sample generalization.

This work analyzes autoencoders to understand feature learning, revealing different phases of learning.

Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.

Constructs a map with prescribed local Lipschitz constants on a subset of a manifold.

Maps between certain Lipschitz manifolds are isometries if they preserve volume.

This paper recovers smooth functions from noisy modulo samples using a three-stage strategy.

Abstract: Lipschitz homeomorphisms are deformed using Perelman's methods.

We compute the local Lipschitz constant of ReLU networks precisely.

Study shows infinite distinct outer metric Lipschitz classes for knots in $S^3$.

The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.

Novikov conjecture reduced to Lipschitz cohomology of groups.

We examine the impact of learning Lipschitz continuous models in the context of model-based reinforcement learning. We provide a novel bound on multi-step prediction error of Lipschitz models where we quantify the error using the Wasserstein metric. We go on to prove an error bound for the value-function estimate arisi…

New MIP formulations for neural network Lipschitz constant estimation.

New method for differentially private optimization with general Lipschitz conditions.

The paper studies properties of intrinsically Lipschitz constants in metric spaces.

New scalable Lipschitz bounds improve neural network robustness analysis.

The paper studies Lipschitz bounds for integral kernels under differentiability assumptions.

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Ru…

Geodesically complete spaces with curvature bounded above have maps with finite energy that are Lipschitz.

We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S^n,Y) are not dense in the Sobolev space W^{1,n}(S^n,Y). On the other hand we show that if a metric space Y is Lipschitz (n-1)-connected, then Lipschitz …

Proves rigidity for maps between manifolds using degree theory and current developments.

Paper investigates Lipschitz constants of self-attention modules in neural networks.